Orientating a
parametrized surfaces
is a little trickier than orienting a curve.
But, if you think of choosing an orientation as choosing between
one of two possible unit vectors, it will help you understand the orientation of
surfaces.

Like curves, we can parametrize a surface in two
different orientations. The orientation of a curve is given by the
unit tangent vector $\vc{n}$; the orientation of a surface is given by
the unit normal vector
$\vc{n}$. Unless we are dealing with an unusual surface, a surface has two sides.
We can pick the normal vector to point out one side of the surface, or
we can pick the normal vector to point out the other side of the
surface. Our choice of normal vector specifies the orientation of the
surface. We call the side of the surface with the normal vector the
positive side of the surface.

As an example, consider the sphere of radius $R$ centered at the
origin. Using spherical coordinates, we could parametrize the
sphere using
\begin{align*}
\dlsp(\theta,\phi) = (R\sin \phi \cos\theta, R\sin\phi \sin\theta, R\cos\phi)
\end{align*}
for $0 \le \theta \le 2\pi$ and $0 \le \phi \le \pi$. Remember that
a formula for the unit normal vector
is
\begin{align*}
\vc{n} = \frac{\displaystyle \pdiff{\dlsp}{\spfv}(\spfv,\spsv) \times
\pdiff{\dlsp}{\spsv}(\spfv,\spsv)}{\displaystyle \left\| \pdiff{\dlsp}{\spfv}(\spfv,\spsv) \times \pdiff{\dlsp}{\spsv}(\spfv,\spsv)
\right\|}.
\end{align*}
only in this case, we have $\theta$ and $\phi$ rather than $\spfv$ and
$\spsv$.

This is the normal vector for the point on the surface given by
$$\dlsp(\theta,\phi) = (R\sin \phi \cos\theta, R\sin\phi \sin\theta,
R\cos\phi).$$ Note that $\vc{n}$ points in the opposite direction as
$\dlsp(\theta,\phi)$ (as $\vc{n}$ is $-1/R$ times
$\dlsp(\theta,\phi)$). Consequently, $\vc{n}$ points back toward the
origin, i.e., points inside the sphere. For example, if $\theta=0$
and $\phi=\pi/2$, then $\dlsp(0,\pi/2) = (R,0,0)$ and
$\vc{n}=(-1,0,0)$. The vector $\vc{n}$ for any value of $\theta$ or
$\phi$ is shown in the figure below. Notice that $\vc{n}$ always
points toward the inside of the sphere. In this case, the inside of
the sphere is the positive side of the surface.

Applet loading

Sphere with inward normal vector. The sphere of a fixed radius $R$ is parametrized by
$\dlsp(\theta,\phi) = (R\sin \phi \cos\theta, R\sin\phi \sin\theta, R\cos\phi)$
for $0 \le \theta \le 2\pi$ and $0 \le \phi \le \pi$. In this case, we have chosen the inward pointing normal vector
$\vc{n} = (-\sin \phi \cos\theta, -\sin\phi \sin\theta, -\cos\phi),$
orienting the surface so the inside is the positive side. You can change the values of $\theta$ and $\phi$ by moving the base of the normal vector or by dragging the points on the sliders.

We could also orient the surface by
choosing the normal vector $\pdiff{\dlsp}{\phi} \times
\pdiff{\dlsp}{\theta}$ instead of the vector $\pdiff{\dlsp}{\theta} \times \pdiff{\dlsp}{\phi}$ that we used above.
How does the new normal vector orient the surface?

Recalling the properties
of the cross product,
we can conclude these vectors are opposites: $\displaystyle
\pdiff{\dlsp}{\phi} \times \pdiff{\dlsp}{\theta} = -\pdiff{\dlsp}{\theta} \times
\pdiff{\dlsp}{\phi}$. In this case, unit normal is
\begin{align*}
\vc{n} = \frac{\displaystyle\pdiff{\dlsp}{\phi} \times \pdiff{\dlsp}{\theta}}
{\displaystyle\left\| \pdiff{\dlsp}{\phi} \times \pdiff{\dlsp}{\theta} \right\|}
= (\sin\phi \cos\theta, \sin\phi \sin\theta,\cos\phi)
\end{align*}
The normal vector points toward the outside the sphere, as shown below. For this second
orientation of the sphere, the outside is the positive side of the
surface.

Applet loading

Sphere with outward normal vector. The sphere of a fixed radius $R$ is parametrized by
$\dlsp(\theta,\phi) = (R\sin \phi \cos\theta, R\sin\phi \sin\theta, R\cos\phi)$
for $0 \le \theta \le 2\pi$ and $0 \le \phi \le \pi$. In this case, we have chosen the outward pointing normal vector
$\vc{n} = (\sin \phi \cos\theta, \sin\phi \sin\theta, \cos\phi),$
orienting the surface so the outside is the positive side. You can change the values of $\theta$ and $\phi$ by moving the base of the normal vector or by dragging the points on the sliders.